Bilinear Forms

## The groups ${U}_{n}$, ${O}_{n}$, $S{p}_{2n}$
The unitary group is $U(n) = g∈GLn(ℂ)∣ g g ‾ t = 1$ where $g ‾ = ( g ‾ ij ) if g=(gij).$ The orthogonal group is $On(ℂ) = g∈GLn(ℂ)∣ g g t = 1 .$ The symplectic group is $Sp2n(ℂ) = g∈GL2n(ℂ)∣ g J g t = 1$ where $J= 1 ⋱ 1 -1 ⋱ -1 or J= 1 ⋰ 1 -1 ⋰ -1$
Let $V$ be an $𝔽$-vector space. A bilinear form on $V$ is a map $⟨,⟩:V×V\to 𝔽$ such that $⟨ c 1 v 1 + c 2 v 2 , v3 ⟩ = c1 ⟨ v1 , v3 ⟩ + c2 ⟨ v2 , v3 ⟩, and ⟨ v 1 , c 1 v 2 + c2 v3 ⟩ = c1 ⟨ v1 , v2 ⟩ + c2 ⟨ v2 , v3 ⟩,$ for ${v}_{1},{v}_{2},{v}_{3}\in V$, ${c}_{1},{c}_{2},{c}_{3}\in 𝔽.$ A bilinear form $⟨,⟩:V×V\to 𝔽$ is symmetric if $⟨ v1 , v2 ⟩ = ⟨ v2 , v1 ⟩ for v1 , v2 ∈ V ,$ and skew-symmetric if $⟨ v1 , v2 ⟩ = - ⟨ v2 , v1 ⟩ for v1 , v2 ∈ V .$ The orthogonal group is $On(𝔽) = O(V,⟨⟩) = g∈GL(V)∣ ⟨ gv1,gv2 ⟩ = ⟨ v1v2 ⟩ for v1,v2∈V .$ The symplectic group is $Spn(𝔽) = O(V,⟨⟩) = g∈GL(V)∣ ⟨ gv1,gv2 ⟩ = ⟨ v1v2 ⟩ for v1,v2∈V .$ Let 𝔽 be a field with an involution